4.1 Laws of Indices
1. Definition and Fundamental Laws
An index (plural: indices) or exponent represents repeated multiplication. For any real numbers $a$ and $b$ where $a, b \neq 0$, and integers $m, n$:
- Multiplication Law: $a^m \times a^n = a^{m+n}$
- Division Law: $a^m \div a^n = a^{m-n}$
- Power Law: $(a^m)^n = a^{mn}$
- Product Power Law: $(ab)^n = a^n b^n$
- Fraction Power Law: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
2. Zero, Negative, and Fractional Indices
- Zero Index: $a^0 = 1 \quad (a \neq 0)$
- Negative Index: $a^{-n} = \frac{1}{a^n}$
- Fractional Indices: $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$
3. Solving Index Equations
Index equations can be solved using two primary approaches:
- Equating Bases: If $a^x = a^y$ and $a \neq 0, \pm 1$, then $x = y$.
- Substitution (Quadratic Form): Equations involving terms like $a^{2x}$ and $a^x$ can be reduced to quadratic form by letting $u = a^x$.
4.2 Laws of Surds
1. Definition of Surds
A surd is an irrational number expressed in root form ($\sqrt[n]{a}$), where $a$ is a rational number and the value cannot be simplified to a exact rational number (e.g., $\sqrt{2}, \sqrt{3}, \sqrt{5}$).
2. Fundamental Rules of Surds
For any positive real numbers $a$ and $b$:
- $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
- $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
- $\sqrt{a} \times \sqrt{a} = (\sqrt{a})^2 = a$
3. Simplifying Surds and Similar Surds
A surd is in its simplest form when the number under the radical sign contains no perfect square factors (e.g., $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$).
Surds with the same irrational factor are called similar surds (e.g., $2\sqrt{3}$ and $5\sqrt{3}$) and can be added or subtracted:
$$p\sqrt{a} \pm q\sqrt{a} = (p \pm q)\sqrt{a}$$
4. Rationalising the Denominator
To eliminate a surd from the denominator of a fraction:
- Single Surd Denominator: Multiply both numerator and denominator by $\sqrt{a}$:
$$\frac{k}{\sqrt{a}} = \frac{k\sqrt{a}}{a}$$
- Binomial Surd Denominator: Multiply numerator and denominator by the conjugate surd:
$$\text{The conjugate of } (\sqrt{a} + \sqrt{b}) \text{ is } (\sqrt{a} - \sqrt{b})$$
$$\frac{k}{\sqrt{a} + \sqrt{b}} = \frac{k(\sqrt{a} - \sqrt{b})}{a - b}$$
4.3 Laws of Logarithms
1. Definition and Relationship with Indices
A logarithm is the inverse operation of exponentiation. If $N = a^x$, then:
$$\log_a N = x$$
where $a > 0$, $a \neq 1$, and $N > 0$.
Important Properties:
- $\log_a 1 = 0$
- $\log_a a = 1$
- $\log_10 N$ is the common logarithm (often written simply as $\lg N$).
- $\log_e N = \ln N$ is the natural logarithm (where $e \approx 2.71828$).
2. Basic Laws of Logarithms
For any positive numbers $a, M, N$ where $a \neq 1$:
- Product Law: $\log_a (MN) = \log_a M + \log_a N$
- Quotient Law: $\log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N$
- Power Law: $\log_a (M^p) = p \log_a M$
3. Change of Base Formula
To change a logarithm from base $a$ to a new base $c$:
$$\log_a b = \frac{\log_c b}{\log_c a}$$
Special reciprocal relation: $\log_a b = \frac{1}{\log_b a}$
4.4 Applications of Indices, Surds and Logarithms
1. Solving Logarithmic and Exponential Equations
- Logarithmic Equations: Combine terms using log laws to get $\log_a f(x) = \log_a g(x) \implies f(x) = g(x)$, or convert to index form $\log_a f(x) = k \implies f(x) = a^k$. Always check for extraneous roots (logarithms of zero or negative numbers are undefined).
- Exponential Equations with Different Bases: Take logarithms on both sides ($a^x = b^y \implies \log a^x = \log b^y \implies x \log a = y \log b$).
2. Modeling Real-World Problems
Logarithms and exponential functions are widely used to model growth, decay, finance, acoustics, and population dynamics (e.g., compound interest $A = P(1+r)^n$, radioactive decay $N(t) = N_0 e^{-\lambda t}$, Richter earthquake scale, pH levels).